let T be non empty TopSpace; :: thesis: for a, b being Element of (Domains_Lattice T)
for A, B being Element of Domains_of T st a = A & b = B holds
( a [= b iff A c= B )
let a, b be Element of (Domains_Lattice T); :: thesis: for A, B being Element of Domains_of T st a = A & b = B holds
( a [= b iff A c= B )
let A, B be Element of Domains_of T; :: thesis: ( a = A & b = B implies ( a [= b iff A c= B ) )
assume that
A1: a = A and
A2: b = B ; :: thesis: ( a [= b iff A c= B )
B in Domains_of T ;
then B in { C where C is Subset of T : C is condensed } by TDLAT_1:def 1;
then A3: ex Q being Subset of T st
( Q = B & Q is condensed ) ;
thus ( a [= b implies A c= B ) :: thesis: ( A c= B implies a [= b )
proof
assume a [= b ; :: thesis: A c= B
then a "\/" b = b by LATTICES:def 3;
then (Int (Cl (A \/ B))) \/ (A \/ B) = B by A1, A2, Th86;
hence A c= B by A3, Th58; :: thesis: verum
end;
assume A c= B ; :: thesis: a [= b
then (Int (Cl (A \/ B))) \/ (A \/ B) = B by A3, Th58;
then a "\/" b = b by A1, A2, Th86;
hence a [= b by LATTICES:def 3; :: thesis: verum